Research Journal of Applied Sciences, Engineering and Technology 5(23): 5408-5412, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: November 24, 2012
Accepted: January 17, 2013
Published: May 28, 2013
Competition between Commercial Open Source Software Firms under the GNU General
Public License
Mingqing Xing
Department of Economics and Management, Weifang University, Weifang 261061, China
Abstract: This study compares R&D incentives for commercial open source software (COSS) firms under the GNU
General Public License (i.e., GPL). It is found that: (i) although the GPL requires firms open the codes of features,
firms have incentives to invest in software features under private optimum; (ii) the firm with high software usability
has much higher incentive to invest in software features, sets higher price, obtains more market share and profit than
the one with low software usability does; (iii) firms invest too little in software features under GPL from a public
policy perspective.
Keywords: Commercial open source software, competition, R&D, general public license, software features,
software usability
INTRODUCTION
Since 1990s, the rapid development of open source
(e.g., Linux) is a significant phenomenon in software
industries. Open Source Software (OSS) is software,
whose sources codes are allowed software developers
to share, identify and correct errors and redistribute
(O’Reilly, 1999). Now more and more companies build
commercial products based on open source software, a
typical example is Red Hat Inc. Commercial Open
Source Software (COSS) is privately developed based
on publicly available source code (Kumar et al., 2011).
A software quality consists of two components:
usability (includes ease of installation, documentation,
user interface and level of technical support) and
features (includes feature set, reliability, security etc)
(Choudhary and Zhou, 2007). By improving the
features or usability of the existing open source
software, firms generate a product that contains both
publicly and privately developed components. The total
amount of work invested into open source software
projects is growing at an exponential rate and can be
expected to continue growing at this rate for a while
before a slowing down (Deshpande and Riehle, 2008).
Firms must follow corresponding open source
license when they develop software based on publicly
available source code. The most common license
dictating how commercial open source software may be
distributed is the GNU General Public License (GPL)
(Laurent, 2004). Under the GPL, firms can freely obtain
the codes of open source software, but the codes must
open when they enhance the software features. This
gives rise to the following issues. First, does a
commercial open source software firm have an
incentive to invest in software features when its
competitors can freely obtain its developments under
the GPL? Second, from a public policy perspective, are
the commercial open source software firms’ R&D
incentives towards software features just the right or too
high (low)? We answer above questions by modifying
the vertical differentiation model (Mussa and Rosen,
1978).
The following works are related to our study.
Raghunathan et al. (2005), Choudhary and Zhou
(2007), Lanzi (2009) and Xing (2010) research the
quality (innovation) competition between open and
closed source software, however they don’t consider
commercial open source and open source licenses. Sen
(2007) investigates the price competition between
commercial version of open source software and
proprietary software. Dixon (2009) and Riehle (2011)
present the core properties of commercial open source
business models and discuss how they work. Although
their papers relates to commercial open source, they
don’t involve in R&D competition.
THE BASIC SETUP
There are three types of open source software
products in a market. One is from the not-for profit
community (called OSS in this study) and the other two
are from the commercial open source software (called
COSS in this study) firms. Firms can freely derive the
codes of software features from the open source
community, but they must comply with the relevant
open source licenses.
Software users are indexed by their level of
technical ability, measured by parameter θ, uniformly
distributed with density 1 over interval [0, 1]. Assume
that users with higher level of technical skills have
lower θ, while those with lower degree of technical
capability have higher θ. Moreover, a user who with
lower technical expertise has higher willingness to pay
for software usability than a user with higher technical
expertise does (Choudhary and Zhou, 2007).
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Res. J. Appl. Sci. Eng. Technol., 5(23): 5408-5412, 2013
A firm must follow the corresponding open source
licenses when develops software based on publicly
available source code. The GNU General Public
License (GPL) is the most common open source
license. This study assumes firms develop the
commercial open source software based on the open
source software of community under the GPL.
The indirect utility functions for the generic
consumer at θ∈ [0, 1] when he/she uses OSS and COSS
are respectively given by:
uo = θ vo + θ vo ( f o + f1 + f 2 ) + ( f o + f1 + f 2 )
(1)
u1 = θ v1 + θ v1 ( f o + f1 + f 2 ) + ( f o + f1 + f 2 ) − p1
(2)
u2 = θ v2 + θ v2 ( f o + f1 + f 2 ) + ( f o + f1 + f 2 ) − p2
(3)
The demand functions for open source community
and firms are respectively given by:
d o = θˆo1 − 0 =
p1
(v1 − vo )(1 + f o + f1 + f 2 )
(8)
d1 = θˆˆ12 − θ o1 =
1
p − p1
p1
( 2
)
−
(1 + f o + f1 + f 2 ) v2 − v1 v1 − vo
(9)
1 θˆ12 =−
1
d 2 =−
p2 − p1
(v2 − v1 )(1 + f o + f1 + f 2 )
(10)
The profit functions for firm 1 and firm 2 are
respectively given by:
=
π 1 p1d1 − γ f12
where, vo , vi is the usability of OSS, COSS i (i = 1,
p1
p − p1
p1
=
−
( 2
) − γ f12
2), satisfies 0 < vo < v1 < v2 ; f o , f i is the initial features
(1 + f o + f1 + f 2 ) v2 − v1 v1 − vo
of OSS and the feature developments of COSS i (i = 1,
2); pi is the price of COSS i (i = 1, 2). Note that:
=
π 2 p2 d 2 − γ f 22
•
•
The price of OSS equals zero (i.e., p o = 0) because
the open source software can be freely available
from the open source community;
This study assumes open source community and
both firms can wholly obtain others’ feature
developments (because the GPL requires firms
open the developments of feature), so all of their
software features equal f o + f1 + f 2 .
The marginal consumer who is indifferent between
using OSS and COSS 1, indexed by θˆo1 , is given by
uo = u1 :
θˆˆo1vo + θ o1vo ( f o + f1 + f 2 ) + ( f o + f1 + f 2 )
= θˆˆo1v1 + θ o1v1 ( f o + f1 + f 2 ) + ( f o + f1 + f 2 ) − p1
p1
(v1 − vo )(1 + f o + f1 + f 2 )
(5)
The marginal consumer who is indifferent between
using COSS 1 and COSS 2, indexed by θˆ12 , is given by
u1 = u2:
θˆˆ12v1 + θ12v1 ( f o + f1 + f 2 ) + ( f o + f1 + f 2 ) − p1
= θˆˆ12v2 + θ12v2 ( f o + f1 + f 2 ) + ( f o + f1 + f 2 ) − p2
Solving (6), we obtain:
θˆ12 =
p2 − p1
(v2 − v1 )(1 + f o + f1 + f 2 )
(12)
where, γ f i 2 denotes the R&D cost when firm i
develops the features of open source software and γ is
positive parameter which measures the innovation
efficiency.
If the commercial open source software firm is run
by a benevolent social planner instead, the levels of
software features are chosen to maximize the social
welfare, defined as the sum of profits and consumer
surplus. The welfare function corresponds to:
SW = π 1 + π 2 + CS
(4)
Solving (4), we obtain:
θˆo1 =
p2 − p1
=
p2 [1 −
] − γ f 22
(v2 − v1 )(1 + f o + f1 + f 2 )
(11)
θˆˆo1
θ12
1
0
θ o1
θ12
= π 1 + π 2 + ∫ uo dθ + ∫ ˆˆ u1dθ + ∫ u2 dθ
(13)
The timing of R&D and price competition is as
follows. In the first stage, firms determine the
developments of software features. In the second stage,
they set price.
THE PRIVATE OPTIMUM
The solutions of the game model are derived by
backwards induction. The price stage is analyzed firstly
and then the R&D stage is decided.
(6)
Solution of Stage 2: The first-order conditions of (11)
and (12) with respect to p 1 and p 2 are respectively
given by:
∂π 1
(7)=
∂p1
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p2 − 2 p1
1
2 p1
−
( =
) 0
v1 − vo
(1 + f o + f1 + f 2 ) v2 − v1
(14)
Res. J. Appl. Sci. Eng. Technol., 5(23): 5408-5412, 2013
∂π 2
2 p2 − p1
2(v2 − vo )(v2 − v1 )
(15)
=
1−
=
0
p2*
=
×
∂p2
(1 + f o + f1 + f 2 )(v2 − v1 )
4(v2 − vo ) − (v1 − vo )

(v2 − v1 )(v2 − vo )[4(v2 − vo ) + (v1 − vo )] 
1 + f o +

2γ [4(v2 − vo ) − (v1 − vo )]2


Solving (14) and (15), we derive the optimal prices
for COSS 1 and COSS 2:
(v1 − vo )(v2 − v1 )(1 + f o + f1 + f 2 )
4(v2 − vo ) − (v1 − vo )
p1 =
p2 =
π 1*
=
(16)
2(v2 − vo )(v2 − v1 )(1 + f o + f1 + f 2 )
4(v2 − vo ) − (v1 − vo )
(v1 − vo )(v2 − v1 )(v2 − vo )
×
[4(v2 − vo ) − (v1 − vo )]2
(v2 − v1 )(v2 − vo )[8(v2 − vo ) + (v1 − vo )] 

1 + f o +

4γ [4(v2 − vo ) − (v1 − vo )]2


(17)
*
v2 − vo
4(v2 − vo ) − (v1 − vo )
(v2 − v1 )(v2 − vo )[2(v2 − vo ) + (v1 − vo )] 

1 + f o +

2γ [4(v2 − vo ) − (v1 − vo )]2


(18)
2(v2 − vo )
d2 =
4(v2 − vo ) − (v1 − vo )
(19)
π1
(v1 − vo )(v2 − v1 )(v2 − vo )(1 + f o + f1 + f 2 )
− γ f12
[4(v2 − vo ) − (v1 − vo )]2
π2
4(v2 − v1 )(v2 − vo ) 2 (1 + f o + f1 + f 2 )
− γ f 22
[4(v2 − vo ) − (v1 − vo )]2
(20)
(28)
4(v − v )(v − v ) 2
o
2
1
2
π2
=
×
Substituting (16) and (17) in (9)-(12), the demand
[4(v2 − vo ) − (v1 − vo )]2
and profit functions for firm 1 and firm 2 are given by:
d1 =
(27)
(29)
According to (18) and (19), the demand functions
don’t depend on f1 and f 2 , so the optimal demands for
firm 1 and firm 2 are given by (18) and (19).
Comparing the optimal results for two firms, we
obtain the following conclusions.
Proposition 1:
(21)
•
Solution of stage 1: Taking the derivatives of (20) and
(21) with respect to f1 and f 2 respectively and setting
them equal to zero, we obtain the following equations:
•
∂π 1 (v1 − vo )(v2 − v1 )(v2 − vo )
=
=
− 2γ f1 0
[4(v2 − vo ) − (v1 − vo )]2
∂f1
∂π 2
4(v2 − v1 )(v2 − vo ) 2
=
=
− 2γ f 2 0
∂f 2 [4(v2 − vo ) − (v1 − vo )]2
(22)
Proof:
f 2* =
2(v2 − v1 )(v2 − vo )
(24)
γ [4(v2 − vo ) − (v1 − vo )]2
(v1 − vo )(v2 − v1 )
×
4(v2 − vo ) − (v1 − vo )
(v2 − v1 )(v2 − vo )[4(v2 − vo ) + (v1 − vo )] 

1 + f o +

2γ [4(v2 − vo ) − (v1 − vo )]2


(v2 − v1 )[2(v2 − vo ) − (v1 − vo )](1 + f o + f1* + f 2* )
>0
4(v2 − vo ) − (v1 − vo )
(25)
because
of v2 − v1 > 0 and 2(v2 − vo ) − (v1 − vo ) > 0 , so p2* > p1*
•
=
d 2* − d1*
2
because
of v2 − v1 > 0 , v2 − vo > 0 and 4(v2 − vo ) − (v1 − vo ) > 0
, so f 2* > f1*
•
=
p2* − p1*
π 2* − π 1*
=
Substituting (24) and (25) in (16), (17), (20) and
(21), the optimal prices and profits for firm 1 and firm 2
are given by:
p1*
=
(v2 − v1 )(v2 − vo )[4(v2 − vo ) − (v1 − vo )]
>0
2γ [4(v2 − vo ) − (v1 − vo )]2
•
=
f 2* − f1*
(23)
Solving (22) and (23), we derive the optimal
improvements of feature for firm 1 and firm 2:
(v − v )(v − v )(v − v )
f1* = 1 o 2 1 2 o 2
2γ [4(v2 − vo ) − (v1 − vo )]
•
Firm 2 invests more in software features than firm
1 does (i.e. f 2* > f1* )
The price of coss 2 is higher than that of coss 1
(i.e., p2* > p1* )
Both demand and profit of firm 1 are more than
that of firm 1 (i.e. d 2* > d1* and π 2* > π 1* ).
(v2 − vo )
>0,
4(v2 − vo ) − (v1 − vo )
so
(v2 − v1 )(v2 − vo )[1 + f o + ( f1* + f 2* ) 2]
>0,
4(v2 − vo ) − (v1 − vo )
d 2* > d1*
so π 2* > π 1*
We use a numerical example to show the results of
v2 2=
v1 4=
vo 1 , γ = 1 and qo = 1 ,
Proposition 1. When =
*
*
*
*
*
there are f 2 = 12 f1 , p2 = 6 p1 , d 2 = 2d1* and π 2* > 11π 1* .
Proposition 1 show that, although the GPL requires
firms open the codes of features, COSS firms have
(26)
incentives to invest in software features. This well
explains why some COSS firms (e.g., Red Hat Inc)
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Res. J. Appl. Sci. Eng. Technol., 5(23): 5408-5412, 2013
make significant contributions to the Linux kernel
Proof:
under the GPL, which implies that they must make
publicly available any feature contributions they makes
1
f 2s − f 2*
=
{(2 + v2 )[4(v2 − vo ) − (v1 − vo )]2
to Linux (Kumar et al., 2011). Moreover, the high4γ [4(v2 − vo ) − (v1 − vo )]2
usability firm has higher incentive to invest in features,
− (v2 − v1 )(v2 − vo )[(v1 − vo ) + 4(v2 − v1 ) + 8(v2 − vo )]} > 0
sets higher price, obtains more market share and profit
than the low-usability one does. For example, Red Hat
Because of
Inc provides users with more documentation,
installation and maintenance and support programs for
(2 + v2 )[4(v2 − vo ) − (v1 − vo )]2
Linux than other COSS firms do (i.e. the usability of
Red Hat Linux is higher than that of other COSS firms).
=(2 + v2 )[16(v2 − v1 ) 2 + 24(v2 − v1 )(v1 − vo ) + 9(v1 − vo ) 2 ]
As a result, Red Hat Inc is more willing to develop the
> 16(v2 − v1 ) 2 v2 + 24(v2 − v1 )(v1 − vo )v2
features of Linux and then sets much higher prices,
> 12(v2 − v1 ) 2 (v2 − vo ) + 9(v2 − v1 )(v1 − vo )(v2 − vo )
obtains much more demand and profit than other COSS
= (v2 − v1 )(v2 − vo )[12(v2 − v1 ) + 9(v1 − vo )]
firms with low usability.
=
(v2 − v1 )(v2 − vo )[(v1 − vo ) + 4(v2 − v1 ) + 8(v2 − vo )]
THE SOCIAL OPTIMUM
We now proceed to decide the levels of software
features make the social welfare maximization. Assume
that the prices are decided by firms and the social
planner only chooses the software features. Combining
(5), (7), (13) and (16)-(21), the social welfare function
is given by:
θˆˆo1
θ12
1
0
θ o1
θ12
SW = π 1 + π 2 + ∫ uo dθ + ∫ ˆˆ u1dθ + ∫ u2 dθ
=
(1 + f o + f1 + f 2 )  (v2 − v1 )(v2 − vo )[(v1 − vo ) + 4(v2 − v1 )] 
v2 −

2
[4(v2 − vo ) − (v1 − vo )]2


2
2
+ ( f o + f1 + f 2 ) − γ ( f1 + f 2 )
(30)
Differentiating (30) with respect to ƒ 1 and ƒ 2
respectively and setting them equal to zero:
∂SW 1 
(v − v )(v − v )[(v − v ) + 4(v2 − v1 )] 
= v2 − 2 1 2 o 1 o

∂f1
2 
[4(v2 − vo ) − (v1 − vo )]2

+ 1 − 2γ f1 =0
(31)
∂SW 1 
(v − v )(v − v )[(v − v ) + 4(v2 − v1 )] 
= v2 − 2 1 2 o 1 o

∂f 2 2 
[4(v2 − vo ) − (v1 − vo )]2

+ 1 − 2γ f 2 =0
(32)
Solving (31) and (32), we derive the feature
improvements under social optimum:
f1s = f 2s =
1 
(v2 − v1 )(v2 − vo )[(v1 − vo ) + 4(v2 − v1 )] 
2 + v2 −

4γ 
[4(v2 − vo ) − (v1 − vo )]2

Therefore, f 2s > f 2* . Moreover, f1s > f1* because of
f1s = f 2s and f 2* > f1* .
We use a numerical example to show the results of
v2 2=
v1 4=
vo 1 , there are
Proposition 2. When =
f1s > 114 f1* and f 2s > 9 f 2* .
Proposition 2 demonstrates that, contrast to the
social optimum, COSS firms invest too little in
software features under private optimum. The results of
Proposition 1 and Proposition 2 indicate that, COSS
firms have incentives to invest in software features
under the GPL, but the innovation levels are much
lower than the planner expects. The reason is that the
GPL requires COSS firms open their any feature
developments, which damages to firm’ innovation
incentives to invest in software features.
CONCLUSION
By extending the vertical differentiation model,
this study investigates the R&D incentives for
commercial open source software firms to invest in
software features in a competitive market. We assume
firms must follow the GNU General Public License
when they develop the software features and find that:
•
(33)
A brief comparative assessment of the two regimes
(profit-seeking firms and social planning) is now in
order. We obtain the following results.
•
•
Proposition 2: the software feature improvements for
both firms under private optimum are less than that
under social optimum.
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The commercial open source software firms have
incentives to invest in software features despite the
GNU General Public License requires firms make
publicly available any feature contributions they
make
The high-usability firm’s R&D incentive towards
software features is higher than that of the lowusability firm
From a public policy perspective, the commercial
open source software firms’ R&D incentive
towards software features is too low under the
GNU General Public License.
Res. J. Appl. Sci. Eng. Technol., 5(23): 5408-5412, 2013
ACKNOWLEDGMENT
We gratefully acknowledge financial support from
Weifang Science and Technology Development Plan
(No.20121105).
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